I’ll see if I have time to write a proper answer later, but if you consider stochastic games a form of control problem, then tug of war games are intimately connected to the $p$-Laplacian, which is an elliptic partial differential operator.
More or less, the $p$-Laplace equation $\nabla_p u = 0$ is the dynamic programming equation for tug of war games. The link above goes into more detail on this connection.